# Topological Optimization of Vehicle ISD Suspension under Steering Braking Condition

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{2}− a

_{1}), where a

_{2}and a

_{1}are the acceleration of the two endpoints, and b is the inertance of the inerter. Together with the spring and damping, it forms the ISD suspension system. At present, vehicles’ passive ISD suspension have been proven to be capable of effectively improving the ride performance of vehicles [19,20], but there are few studies on the application of the inerter to improve the anti-roll performance of vehicles [21]. It can be seen that the inerter has great potential in improving the anti-roll performance of vehicles. Shen Yujie solves the problem of low-frequency and high-frequency vibration suppression of vehicles by designing a passive fractional order electrical network [22] and PDD-based active control design of electromechanical ISD suspension [23]. Literature [21] selected passive suspension, the suspension of a damper and an inerter in parallel, the suspension of a damper and an inerter in series, and three-element suspension based on the dynamic vibration absorption principle (all supporting springs in parallel) built a seven-degree of freedom vehicle model and analyzed the influence of four suspension structures on vehicle roll. On this basis, a 14-degree-of-freedom model considering wheel nonlinearity is established, and a vehicle acceleration braking model is added. The influence of ISD suspension structure on vehicle pitch motion is studied. According to the research results of the literature [21], the improved three-element topological inertial suspension structure with a supporting spring in parallel is helpful in improving the roll stability of the vehicle. Therefore, this paper comprehensively considers eight kinds of improved three-element topological suspension structures with supporting springs in parallel, adds braking conditions, and analyzes their comprehensive influence on the pitch stability, roll stability, and vertical stability of the vehicle.

## 3. Construction of Vehicle Nonlinear Dynamics Model

#### 3.1. Vehicle Dynamics Model

- (1)
- Ignore the impact of air resistance and rolling resistance on the car;
- (2)
- Ignore the influence of the steering system and assume that the left and right front wheel angles are the same as the input variables;
- (3)
- Assume that the center of gravity of the car coincides with the origin of the moving coordinate system when the car is driving;
- (4)
- Assume that the four tires have the same characteristics.

#### 3.2. Tire Model

#### 3.3. ISD Suspension Model

## 4. Action Law of ISD Suspension Topology

#### 4.1. Pavement Input Model

^{3}.

#### 4.2. The Influence of Eight ISD Suspension Structures on Vehicle Performance

## 5. Optimization of Vehicle ISD Suspension Parameters Based on NSGA-II

#### 5.1. Optimal Target Selection

#### 5.2. Constraint Selection

#### 5.3. NSGA-Ⅱ Algorithm Steps

^{2}), so the time complexity of M targets is O(MN

^{2}). In the second round of sorting, an individual i is removed from the non-dominated solution set of the previous round, and then an individual j is removed from ${S}_{I}$. At this time, ${n}_{i}-1$ is added to the non-dominated solution set of the next round. If ${n}_{i}$ = 0, this individual j is added to the non-dominated solution set. Therefore, the total time complexity of non-dominated sorting is O(MN

^{2}) = O(MN

^{2}) + O(N

^{2}).

#### 5.4. Optimization Result

## 6. Simulation Analysis

#### 6.1. Load Transfer Rate

#### 6.2. Step Steering Braking

#### 6.3. Fishhook Steering Braking

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Qin, Y.; Zhao, Z.; Wang, Z.; Li, G. Study of Longitudinal—Vertical Dynamics for In-Wheel Motor-Driven Electric Vehicles. Automot. Innov.
**2021**, 4, 227–237. [Google Scholar] [CrossRef] - Zhao, D.; Yin, Y.; Ni, T.; Zhang, W. Research review on Chassis Integrated Control of Heavy Truck. J. Yanshan Univ.
**2020**, 44, 189–197. [Google Scholar] - Pan, G.; Ding, C.; Wang, W.; Li, A. Application of extended zero moment point in vehicle pitch control evaluation and Active control. J. Jiangsu Univ.
**2021**, 42, 85–91. [Google Scholar] - Jiang, Z.; Zheng, M.; Zhang, N. Research on Pitch dynamics of semi-active anti-pitch Hydraulic Interconnected Suspension. J. Vib. Shock.
**2020**, 39, 272–278. [Google Scholar] - Wu, X.; Zhou, B.; Wen, G. Research on Anti-roll Control of Hydraulic Interconnected Suspension. China J. Highw. Transp.
**2018**, 31, 123–132. [Google Scholar] - Zhang, L.; Meng, Q.; Chen, H.; Huang, Y.; Liu, Y.; Guo, K. Kalman Filter-Based Fusion Estimation Method of Steering Feedback Torque for Steer-by-Wire Systems. Automot. Innov.
**2021**, 4, 430–439. [Google Scholar] [CrossRef] - Qin, Y.; Wang, Z.; Yuan, K.; Zhang, Y. Correction to: Comprehensive Analysis and Optimization of Dynamic Vibration-Absorbing Structures for Electric Vehicles Driven by In-Wheel Motors. Automot. Innov.
**2020**, 3, 192. [Google Scholar] [CrossRef] - Wang, J.; Liu, Z.; Meng, L.; Fu, T.; Li, J.; Wang, Z. Design and Research of Electro-Hydraulic Suspension System with both Energy Feed and active anti-roll Function. J. Automot. Eng.
**2023**, 13, 396–407. [Google Scholar] - Hou, X.; Zhang, J.; Liu, W.; Ji, Y. LuGre Mode-based Longitudinal Ride Comfort Control of Vehicle during the Post-braking Phase. In Proceedings of the 2020 Chinese Automation Congress (CAC), Shanghai, China, 6–8 November 2020; IEEE: Piscataway, NJ, USA, 2020. [Google Scholar]
- Xu, Z.; Liu, H.; Dang, W.; Hu, C.; Long, Y. Research Review of Automobile Brake Comfort Control. Sci. Technol. Eng.
**2022**, 22, 6790–6801. [Google Scholar] - Bi, H.; Lu, F.; Duan, S.; Huang, M.; Zhu, J.; Liu, M. Two-level principal–agent model for schedule risk control of IT outsourcing project based on genetic algorithm. Eng. Appl. Artif. Intell.
**2020**, 91, 103584. [Google Scholar] [CrossRef] - Wen, H.; Wang, S.; Lu, F.; Feng, M.; Wang, L.; Xiong, J.; Si, M. Colony search optimization algorithm using global optimization. J. Supercomput.
**2022**, 78, 6567–6611. [Google Scholar] [CrossRef] - Li, S.; Du, P.; Feng, X. Integrated Control Strategy of anti-rollover Chassis for Minibus. J. Jiangsu Univ.
**2022**, 43, 131–138. [Google Scholar] - Zhao, S.; Li, Y.; Yu, Q. Vehicle Stability Control Based on Coordinated Control of Multiple Sub-systems of Chassis. J. Traffic Transp. Eng.
**2015**, 15, 77–85. [Google Scholar] - Chen, L.; Chen, M.; Sun, X.; Cai, Y.; Pak, K.W.; Wu, Z. Lateral Dynamics Modeling and Stability Analysis of Air Suspension Bus. Automot. Eng.
**2022**, 44, 1746–1754. [Google Scholar] - Huang, K.; Zhao, P.; Chen, J. Research on Improving Performance of Hydraulic Interconnected Suspension with Inertial Vessel. Noise Vib. Control.
**2022**, 42, 17–22. [Google Scholar] - Lu, S.; Li, Y.; Zheng, L. Research on Vehicle active Rollover Control based on Brake and Suspension System. Automot. Eng.
**2011**, 33, 669–675. [Google Scholar] - Smith, M.C. Synthesis of Networks: The inerter. IEEE Trans. Autom. Control.
**2002**, 47, 1648–1662. [Google Scholar] [CrossRef] - Chen, L.; Yang, X.; Wang, R.; Huang, C.; Shen, Y. Study on the Performance of Improved ISD Three-Component Passive Suspension. Automot. Eng.
**2014**, 36, 340–345. [Google Scholar] - Ge, Z.; Wang, W.; Li, G.; Rao, D. Design, Parameter Optimisation, and Performance Analysis of Active Tuned Inerter Damper (TID) Suspension for Vehicle. J. Sound Vib.
**2022**, 525, 116750. [Google Scholar] [CrossRef] - Shen, Y.; Chen, L.; Liu, Y.; Zhang, X.; Yang, X. Improvement of the lateral stability of vehicle suspension incorporating inerter. Sci. China
**2018**, 61, 1244–1252. [Google Scholar] [CrossRef] - Shen, Y.; Hua, J.; Fan, W.; Liu, Y.; Yang, X.; Chen, L. Optimal design and dynamic performance analysis of a fractional-order electrical network-based vehicle mechatronic ISD suspension. Mech. Syst. Signal Process.
**2023**, 184, 109718. [Google Scholar] [CrossRef] - Shen, Y.; Jia, M.; Yang, X.; Liu, Y.; Chen, L. Vibration suppression using a mechatronic PDD-ISD-combined vehicle suspension system. Int. J. Mech. Sci.
**2023**, 250, 108277. [Google Scholar] [CrossRef] - Luo, J.; Li, P.; Li, P.; Cai, Q. Observer-based multi-objective integrated control for vehicle lateral stability and active suspension design. J. Sound Vib.
**2021**, 508, 116222. [Google Scholar] [CrossRef] - Feng, S.; Zhao, Y.; Deng, H.; Wang, Q.; Chen, T. Parameter Identification of Magic Formula Tire Model Based on Fibonacci Tree Optimization Algorithm. J. Shanghai Jiao Tong Univ.
**2021**, 26, 647–657. [Google Scholar] [CrossRef] - Meng, L. Research on Active Anti-Roll Control Performance of Automotive Electro-Hydraulic Suspension. Ph.D. Dissertation, Jilin University, Changchun, China, 2023. [Google Scholar]
- Qi, Q.; Wu, T. Multi-objective Production Intelligent Scheduling Based on Improved NSGA-II Algorithm. Comput. Technol. Dev.
**2021**, 31, 162–168. [Google Scholar]

**Figure 5.**The variation rule of the RMS value of vehicle body acceleration. (

**a**) The vehicle body acceleration law of L1, L4, L5, and L6; (

**b**) the vehicle body acceleration law of L2, L3, L7, and L8; (

**c**) local amplification of the vehicle body acceleration of L1, L4, L5, L6, and L8.

**Figure 6.**The variation law of the RMS value of roll angle acceleration. (

**a**) The roll angle acceleration law of L1, L4, L5, and L6; (

**b**) the roll angle acceleration law of L2, L3, L7, and L8; (

**c**) local amplification of the roll angle acceleration of L1, L4, L5, L6, and L8.

**Figure 7.**The variation rule of the RMS value of the pitch angle acceleration. (

**a**) The pitch angle acceleration law of L1, L4, L5, and L6; (

**b**) the pitch angle acceleration law of L2, L3, L7, and L8; (

**c**) local amplification of the pitch angle acceleration of L1, L4, L5, L6, and L8.

**Figure 8.**The variation rule of RMS value of suspension working space. (

**a**) The suspension working space law of L1, L4, L5, and L6; (

**b**) the suspension working space law of L2, L3, L7, and L8; (

**c**) local amplification of the suspension working space of L1, L4, L5, L6, and L8.

**Figure 9.**The variation rule of RMS value of dynamic tire load. (

**a**) The dynamic tire load law of L1, L4, L5, and L6; (

**b**) the dynamic tire load law of L2, L3, L7, and L8; (

**c**) local amplification of the dynamic tire load of L1, L4, L5, L6, and L8.

**Figure 11.**Body acceleration under step steering braking. (

**a**) Time domain diagram of body acceleration; (

**b**) frequency domain diagram of body acceleration.

**Figure 12.**Roll angle acceleration under step steering braking. (

**a**) Time domain diagram of roll angle acceleration; (

**b**) frequency domain diagram of roll angle acceleration.

**Figure 13.**Pitch angle acceleration under step steering braking. (

**a**) Time domain diagram of pitch angle acceleration; (

**b**) frequency domain diagram of pitch angle acceleration.

Parameter | Value |
---|---|

Vehicle mass m_{t}/kg | 1659 |

Sprung mass m_{s}/kg | 1410 |

Front wheel unsprung mass m_{ufl},m_{ufr}/kg | 26.5 |

Rear wheel unsprung mass m_{url},m_{urr}/kg | 24.4 |

Left wheel base w_{f}/m | 1.574 |

Left and right wheel base w_{r}/m | 1.593 |

Distance from front axle to center of mass l_{f}/m | 1.278 |

Distance from rear axis to center of mass l_{r}/m | 1.430 |

Height of center of mass h_{1}/m | 0.50 |

Roll height h_{2}/m | 0.40 |

Distance from center of roll to center of mass h_{3}/m | 0.25 |

Body roll moment of inertia I_{x}/kg·m^{2} | 925 |

Body pitch moment of inertia I_{y}/kg·m^{2} | 2577 |

Body yaw moment of inertia I_{z}/kg·m^{2} | 2603 |

Wheel inertia I_{w}/kg·m^{2} | 0.99 |

Wheel radius R_{w}/m | 0.345 |

Front suspension spring stiffness of the original model k_{f0}/kN·m^{−1} | 25 |

Rear suspension spring stiffness of the original model k_{r0}/kN·m^{−1} | 22 |

Front suspension damping coefficient of the original model c_{f0}/N·s·m^{−1} | 1800 |

Rear suspension damping coefficient of the original model c_{r0}/N·s·m^{−1} | 1500 |

Tire stiffness k_{t}/kN·m^{−1} | 192 |

Front suspension roll stiffness k_{1}/N·m·rad^{−1} | 47,298 |

Rear suspension roll stiffness k_{2}/N·m·rad^{−1} | 37,311 |

Performance Index | RMS Value |
---|---|

Body acceleration/(m·s^{−2}) | 0.8684 |

Roll angle acceleration/(rad·s^{−2}) | 0.6338 |

Pitch angle acceleration/(rad·s^{−2}) | 0.4885 |

Suspension working space/(m) | 0.0395 |

Dynamic tire load/(KN) | 0.8771 |

Suspension Structure | ${\mathit{k}}_{\mathit{f}}$ (N/m) | ${\mathit{k}}_{\mathit{r}}$ (N/m) | ${\mathit{c}}_{\mathit{f}}$ (N∙s/m) | ${\mathit{c}}_{\mathit{r}}$ (N∙s/m) | ${\mathit{b}}_{\mathit{f}}$ (kg) | ${\mathit{b}}_{\mathit{r}}$ (kg) |
---|---|---|---|---|---|---|

L4 | 11,395 | 17,269 | 2213 | 2895 | 376 | 4514 |

L5 | 28,708 | 8310 | 2860 | 3406 | 1023 | 3484 |

L6 | / | / | 2815 | 3425 | 3168 | 3614 |

L8 | 8583 | 16,313 | 2971 | 2421 | 102 | 2180 |

RMS Value | L4 | Improvement | L5 | Improvement | L6 | Improvement | L8 | Improvement |
---|---|---|---|---|---|---|---|---|

Body acceleration/(m·s^{−2}) | 0.6542 | 24.67% | 0.6089 | 29.88% | 0.6230 | 27.51% | 0.5604 | 35.47% |

Roll angle acceleration/(rad·s^{−2}) | 0.5972 | 12.61% | 0.6126 | 10.35% | 0.6107 | 10.64% | 0.5859 | 14.26% |

Pitch angle acceleration/(rad·s^{−2}) | 0.3951 | 19.12% | 0.3776 | 10.35% | 0.4077 | 16.54% | 0.4567 | 6.52% |

Suspension working space/(m) | 0.0389 | 1.44% | 0.0383 | 2.78% | 0.0394 | 0.20% | 0.0368 | 6.92% |

Dynamic tire load/(kN) | 0.7188 | 18.05% | 0.8033 | 8.42% | 0.7905 | 9.88% | 0.7465 | 14.90% |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Shi, D.; Du, F.; Yang, X.; Zhu, K.
Topological Optimization of Vehicle ISD Suspension under Steering Braking Condition. *World Electr. Veh. J.* **2023**, *14*, 297.
https://doi.org/10.3390/wevj14100297

**AMA Style**

Liu Y, Shi D, Du F, Yang X, Zhu K.
Topological Optimization of Vehicle ISD Suspension under Steering Braking Condition. *World Electric Vehicle Journal*. 2023; 14(10):297.
https://doi.org/10.3390/wevj14100297

**Chicago/Turabian Style**

Liu, Yanling, Dongyin Shi, Fu Du, Xiaofeng Yang, and Kerong Zhu.
2023. "Topological Optimization of Vehicle ISD Suspension under Steering Braking Condition" *World Electric Vehicle Journal* 14, no. 10: 297.
https://doi.org/10.3390/wevj14100297